Let G be a finite group. The intersection graph of subgroups of G is a graph whose vertices are all non-trivial subgroups of G and in which two distinct vertices H and K are adjacent if and only if H∩K≠1. In this paper, we classify all finite abelian groups whose thickness and outerthickness of subgroup intersection graphs are 1 and 2, respectively. We also investigate the thickness and outerthickness of subgroup intersection graphs for some finite non-abelian groups.
Citation: Huadong Su, Ling Zhu. Thickness of the subgroup intersection graph of a finite group[J]. AIMS Mathematics, 2021, 6(3): 2590-2606. doi: 10.3934/math.2021157
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Let G be a finite group. The intersection graph of subgroups of G is a graph whose vertices are all non-trivial subgroups of G and in which two distinct vertices H and K are adjacent if and only if H∩K≠1. In this paper, we classify all finite abelian groups whose thickness and outerthickness of subgroup intersection graphs are 1 and 2, respectively. We also investigate the thickness and outerthickness of subgroup intersection graphs for some finite non-abelian groups.
Let W be a set and H:W⟶W be a mapping. A point w∈W is called a fixed point of H if w=Hw. Fixed point theory plays a fundamental role in functional analysis (see [15]). Shoaib [17] introduced the concept of α-dominated mapping and obtained some fixed point results (see also [1,2]). George et al. [11] introduced a new space and called it rectangular b-metric space (r.b.m. space). The triangle inequality in the b-metric space was replaced by rectangle inequality. Useful results on r.b.m. spaces can be seen in ([5,6,8,9,10]). Ćirić introduced new types of contraction and proved some metrical fixed point results (see [4]). In this article, we introduce Ćirić type rational contractions for α -dominated mappings in r.b.m. spaces and proved some metrical fixed point results. New interesting results in metric spaces, rectangular metric spaces and b-metric spaces can be obtained as applications of our results.
Definition 1.1. [11] Let U be a nonempty set. A function dlb:U×U→[0,∞) is said to be a rectangular b-metric if there exists b≥1 such that
(ⅰ) dlb(θ,ν)=dlb(ν,θ);
(ⅱ) dlb(θ,ν)=0 if and only if θ=ν;
(ⅲ) dlb(θ,ν)≤b[dlb(θ,q)+dlb(q,l)+dlb(l,ν)] for all θ,ν∈U and all distinct points q,l∈U∖{θ,ν}.
The pair (U,dlb) is said a rectangular b-metric space (in short, r.b.m. space) with coefficient b.
Definition 1.2. [11] Let (U,dlb) be an r.b.m. space with coefficient b.
(ⅰ) A sequence {θn} in (U,dlb) is said to be Cauchy sequence if for each ε>0, there corresponds n0∈N such that for all n,m≥n0 we have dlb(θm,θn)<ε or limn,m→+∞dlb(θn,θm)=0.
(ⅱ) A sequence {θn} is rectangular b-convergent (for short, (dlb)-converges) to θ if limn→+∞dlb(θn,θ)=0. In this case θ is called a (dlb)-limit of {θn}.
(ⅲ) (U,dlb) is complete if every Cauchy sequence in Udlb-converges to a point θ∈U.
Let ϖb, where b≥1, denote the family of all nondecreasing functions δb:[0,+∞)→[0,+∞) such that ∑+∞k=1bkδkb(t)<+∞ and bδb(t)<t for all t>0, where δkb is the kth iterate of δb. Also bn+1δn+1b(t)=bnbδb(δnb(t))<bnδnb(t).
Example 1.3. [11] Let U=N. Define dlb:U×U→R+∪{0} such that dlb(u,v)=dlb(v,u) for all u,v∈U and α>0
dlb(u,v)={0, if u=v;10α, if u=1, v=2;α, if u∈{1,2} and v∈{3};2α, if u∈{1,2,3} and v∈{4};3α, if u or v∉{1,2,3,4} and u≠v. |
Then (U,dlb) is an r.b.m. space with b=2>1. Note that
d(1,4)+d(4,3)+d(3,2)=5α<10α=d(1,2). |
Thus dlb is not a rectangular metric.
Definition 1.4. [17] Let (U,dlb) be an r.b.m. space with coefficient b. Let S:U→U be a mapping and α:U×U→[0,+∞). If A⊆U, we say that the S is α-dominated on A, whenever α(i,Si)≥1 for all i∈A. If A=U, we say that S is α-dominated.
For θ,ν∈U, a>0, we define Dlb(θ,ν) as
Dlb(θ,ν)=max{dlb(θ,ν),dlb(θ,Sθ).dlb(ν,Sν)a+dlb(θ,ν),dlb(θ,Sθ),dlb(ν,Sν)}. |
Now, we present our main result.
Theorem 2.1. Let (U,dlb) be a complete r.b.m. space with coefficient b, α:U×U→[0,∞),S:U→U, {θn} be a Picard sequence and S be a α-dominated mapping on {θn}. Suppose that, for some δb∈ϖb, we have
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)), | (2.1) |
for all θ,ν∈{θn} with α(θ,ν)≥1. Then {θn} converges to θ∗∈U. Also, if (2.1) holds for θ∗ and α(θn,θ∗)≥1 for all n∈N∪{0}, then S has a fixed point θ∗ in U.
Proof. Let θ0∈U be arbitrary. Define the sequence {θn} by θn+1=Sθn for all n∈N∪{0}. We shall show that {θn} is a Cauchy sequence. If θn=θn+1, for some n∈N, then θn is a fixed point of S. So, suppose that any two consecutive terms of the sequence are not equal. Since S:U→U be an α-dominated mapping on {θn}, α(θn,Sθn)≥1 for all n∈N∪{0} and then α(θn,θn+1)≥1 for all n∈N∪{0}. Now by using inequality (2.1), we obtain
dlb(θn+1,θn+2)=dlb(Sθn,Sθn+1)≤δb(Dlb(θn,θn+1))≤δb(max{dlb(θn,θn+1),dlb(θn,θn+1).dlb(θn+1,θn+2)a+dlb(θn,θn+1),dlb(θn,θn+1),dlb(θn+1,θn+2)})≤δb(max{dlb(θn,θn+1),dlb(θn+1,θn+2)}). |
If max{dlb(θn,θn+1),dlb(θn+1,θn+2)}=dlb(θn+1,θn+2), then
dlb(θn+1,θn+2)≤δb(dlb(θn+1,θn+2))≤bδb(dlb(θn+1,θn+2)). |
This is the contradiction to the fact that bδb(t)<t for all t>0. So
max{dlb(θn,θn+1),dlb(θn+1,θn+2)}=dlb(θn,θn+1). |
Hence, we obtain
dlb(θn+1,θn+2)≤δb(dlb(θn,θn+1))≤δ2b(dlb(θn−1,θn)) |
Continuing in this way, we obtain
dlb(θn+1,θn+2)≤δn+1b(dlb(θ0,θ1)). | (2.2) |
Suppose for some n,m∈N with m>n, we have θn=θm. Then by (2.2)
dlb(θn,θn+1)=dlb(θn,Sθn)=dlb(θm,Sθm)=dlb(θm,θm+1)≤δm−nb(dlb(θn,θn+1))<bδb(dlb(θn,θn+1)) |
As dlb(θn,θn+1)>0, so this is not true, because bδb(t)<t for all t>0. Therefore, θn≠θm for any n,m∈N. Since ∑+∞k=1bkδkb(t)<+∞, for some ν∈N, the series ∑+∞k=1bkδkb(δν−1b(dlb(θ0,θ1))) converges. As bδb(t)<t, so
bn+1δn+1b(δν−1b(dlb(θ0,θ1)))<bnδnb(δν−1b(dlb(θ0,θ1))), for all n∈N. |
Fix ε>0. Then ε2=ε′>0. For ε′, there exists ν(ε′)∈N such that
bδb(δν(ε′)−1b(dlb(θ0,θ1)))+b2δ2b(δν(ε′)−1b(dlb(θ0,θ1)))+⋯<ε′ | (2.3) |
Now, we suppose that any two terms of the sequence {θn} are not equal. Let n,m∈N with m>n>ν(ε′). Now, if m>n+2,
dlb(θn,θm)≤b[dlb(θn,θn+1)+dlb(θn+1,θn+2)+dlb(θn+2,θm)]≤b[dlb(θn,θn+1)+dlb(θn+1,θn+2)]+b2[dlb(θn+2,θn+3)+dlb(θn+3,θn+4)+dlb(θn+4,θm)]≤b[δnb(dlb(θ0,θ1))+δn+1b(dlb(θ0,θ1))]+b2[δn+2b(dlb(θ0,θ1))+δn+3b(dlb(θ0,θ1))]+b3[δn+4b(dlb(θ0,θ1))+δn+5b(dlb(θ0,θ1))]+⋯≤bδnb(dlb(θ0,θ1))+b2δn+1b(dlb(θ0,θ1))+b3δn+2b(dlb(θ0,θ1))+⋯=bδb(δn−1b(dlb(θ0,θ1)))+b2δ2b(δn−1b(dlb(θ0,θ1)))+⋯. |
By using (2.3), we have
dlb(θn,θm)<bδb(δν(ε′)−1b(dlb(θ0,θ1)))+b2δ2b(δν(ε′)−1b(dlb(θ0,θ1)))+⋯<ε′<ε. |
Now, if m=n+2, then we obtain
dlb(θn,θn+2)≤b[dlb(θn,θn+1)+dlb(θn+1,θn+3)+dlb(θn+3,θn+2)]≤b[dlb(θn,θn+1)+b[dlb(θn+1,θn+2)+dlb(θn+2,θn+4)+dlb(θn+4,θn+3)]+dlb(θn+3,θn+2)]≤bdlb(θn,θn+1)+b2dlb(θn+1,θn+2)+bdlb(θn+2,θn+3)+b2dlb(θn+3,θn+4)+b3[dlb(θn+2,θn+3)+dlb(θn+3,θn+5)+dlb(θn+5,θn+4)]≤bdlb(θn,θn+1)+b2dlb(θn+1,θn+2)+(b+b3)dlb(θn+2,θn+3)+b2dlb(θn+3,θn+4)+b3dlb(θn+5,θn+4)+b4[dlb(θn+3,θn+4)+dlb(θn+4,θn+6)+dlb(θn+6,θn+5)]≤bdlb(θn,θn+1)+b2dlb(θn+1,θn+2)+(b+b3)dlb(θn+2,θn+3)+(b2+b4)dlb(θn+3,θn+4)+b3dlb(θn+5,θn+4)+b4dlb(θn+6,θn+5)+b5[dlb(θn+4,θn+5)+dlb(θn+5,θn+7)+dlb(θn+7,θn+6)]≤bdlb(θn,θn+1)+b2dlb(θn+1,θn+2)+(b+b3)dlb(θn+2,θn+3)+(b2+b4)dlb(θn+3,θn+4)+(b3+b5)dlb(θn+4,θn+5)+⋯<2[bdlb(θn,θn+1)+b2dlb(θn+1,θn+2)+b3dlb(θn+2,θn+3)+b4dlb(θn+3,θn+4)+b5dlb(θn+4,θn+5)+⋯]≤2[bδnb(dlb(θ0,θ1))+b2δn+1b(dlb(θ0,θ1))+b3δn+2b(dlb(θ0,θ1))+⋯]<2[bδb(δν(ε′)−1b(dlb(θ0,θ1)))+b2δ2b(δν(ε′)−1b(dlb(θ0,θ1)))+⋯]<2ε′=ε. |
It follows that
limn,m→+∞dlb(θn,θm)=0. | (2.4) |
Thus {θn} is a Cauchy sequence in (U,dlb). As (U,dlb) is complete, so there exists θ∗ in U such that {θn} converges to θ∗, that is,
limn→+∞dlb(θn,θ∗)=0. | (2.5) |
Now, suppose that dlb(θ∗,Sθ∗)>0. Then
dlb(θ∗,Sθ∗)≤b[dlb(θ∗,θn)+dlb(θn,θn+1)+dlb(θn+1,Sθ∗)≤b[dlb(θ∗,θn+1)+dlb(θn,θn+1)+dlb(Sθn,Sθ∗). |
Since α(θn,θ∗)≥1, we obtain
dlb(θ∗,Sθ∗)≤bdlb(θ∗,θn+1)+bdlb(θn,θn+1)+bδb(max{dlb(θn,θ∗),dlb(θ∗,Sθ∗).dlb(θn,θn+1)a+dlb(θn,θ∗), dlb(θn,θn+1) dlb(θ∗,Sθ∗)}). |
Letting n→+∞, and using the inequalities (2.4) and (2.5), we obtain dlb(θ∗,Sθ∗)≤bδb(dlb(θ∗,Sθ∗)). This is not true, because bδb(t)<t for all t>0 and hence dlb(θ∗,Sθ∗)=0 or θ∗=Sθ∗. Hence S has a fixed point θ∗ in U.
Remark 2.2. By taking fourteen different proper subsets of Dlb(θ,ν), we can obtainvnew results as corollaries of our result in a complete r.b.m. space with coefficient b.
We have the following result without using α-dominated mapping.
Theorem 2.3. Let (U,dlb) be a complete r.b.m. space with coefficient b,S:U→U, {θn} be a Picard sequence. Suppose that, for some δb∈ϖb, we have
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)) | (2.6) |
for all θ,ν∈{θn}. Then {θn} converges to θ∗∈U. Also, if (2.6) holds for θ∗, then S has a fixed point θ∗ in U.
We have the following result by taking δb(t)=ct, t∈R+ with 0<c<1b without using α-dominated mapping.
Theorem 2.4. Let (U,dlb) be a complete r.b.m. space with coefficient b, S:U→U, {θn} be a Picard sequence. Suppose that, for some 0<c<1b, we have
dlb(Sθ,Sν)≤c(Dlb(θ,ν)) | (2.7) |
for all θ,ν∈{θn}. Then {θn} converges to θ∗∈U. Also, if (2.7) holds for θ∗, then S has a fixed point θ∗ in U.
Ran and Reurings [16] gave an extension to the results in fixed point theory and obtained results in partially ordered metric spaces. Arshad et al. [3] introduced ⪯-dominated mappings and established some results in an ordered complete dislocated metric space. We apply our result to obtain results in ordered complete r.b.m. space.
Definition 2.5. (U,⪯,dlb) is said to be an ordered complete r.b.m. space with coefficient b if
(ⅰ) (U,⪯) is a partially ordered set.
(ⅱ) (U,dlb) is an r.b.m. space.
Definition 2.6. [3] Let U be a nonempty set, ⪯ is a partial order on θ. A mapping S:U→U is said to be ⪯-dominated on A if a⪯Sa for each a∈A⊆θ. If A=U, then S:U→U is said to be ⪯-dominated.
We have the following result for ⪯-dominated mappings in an ordered complete r.b.m. space with coefficient b.
Theorem 2.7. Let (U,⪯,dlb) be an ordered complete r.b.m. space with coefficient b, S:U→U,{θn} be a Picard sequence and S be a ⪯-dominated mapping on {θn}. Suppose that, for some δb∈ϖb, we have
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)), | (2.8) |
for all θ,ν∈{θn} with θ⪯ν. Then {θn} converges to θ∗∈U. Also, if (2.8) holds for θ∗ and θn⪯θ∗ for all n∈N∪{0}. Then S has a fixed point θ∗ in U.
Proof. Let α:U×U→[0,+∞) be a mapping defined by α(θ,ν)=1 for all θ,ν∈U with θ⪯ν and α(θ,ν)=411 for all other elements θ,ν∈U. As S is the dominated mappings on {θn}, so θ⪯Sθ for all θ∈{θn}. This implies that α(θ,Sθ)=1 for all θ∈{θn}. So S:U→U is the α-dominated mapping on {θn}. Moreover, inequality (2.8) can be written as
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)) |
for all elements θ,ν in {θn} with α(θ,ν)≥1. Then, as in Theorem 2.1, {θn} converges to θ∗∈U. Now, θn⪯θ∗ implies α(θn,θ∗)≥1. So all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, S has a fixed point θ∗ in U.
Now, we present an example of our main result. Note that the results of George et al. [11] and all other results in rectangular b-metric space are not applicable to ensure the existence of the fixed point of the mapping given in the following example.
Example 2.8. Let U=A∪B, where A={1n:n∈{2,3,4,5}} and B=[1,∞]. Define dl:U×U→[0,∞) such that dl(θ,ν)=dl(ν,θ) for θ,ν∈U and
{dl(12,13)=dl(14,15)=0.03dl(12,15)=dl(13,14)=0.02dl(12,14)=dl(15,13)=0.6dl(θ,ν)=|θ−ν|2 otherwise |
be a complete r.b.m. space with coefficient b=4>1 but (U,dl) is neither a metric space nor a rectangular metric space. Take δb(t)=t10, then bδb(t)<t. Let S:U→U be defined as
Sθ={15 ifθ∈A13 ifθ=19θ100+85 otherwise. |
Let θ0=1. Then the Picard sequence {θn} is {1,13,15,15,15,⋯}. Define
α(θ,ν)={85 ifθ,ν∈{θn}47 otherwise. |
Then S is an α-dominated mapping on {θn}. Now, S satisfies all the conditions of Theorem 2.1. Here 15 is the fixed point in U.
Jachymski [13] proved the contraction principle for mappings on a metric space with a graph. Let (U,d) be a metric space and △ represents the diagonal of the cartesian product U×U. Suppose that G be a directed graph having the vertices set V(G) along with U, and the set E(G) denoted the edges of U included all loops, i.e., E(G)⊇△. If G has no parallel edges, then we can unify G with pair (V(G),E(G)). If l and m are the vertices in a graph G, then a path in G from l to m of length N(N∈N) is a sequence {θi}Ni=o of N+1 vertices such that lo=l,lN=m and (ln−1,ln)∈E(G) where i=1,2,⋯N (see for detail [7,8,12,14,18,19]). Recently, Younis et al. [20] introduced the notion of graphical rectangular b-metric spaces (see also [5,6,21]). Now, we present our result in this direction.
Definition 3.1. Let θ be a nonempty set and G=(V(G),E(G)) be a graph such that V(G)=U and A⊆U. A mapping S:U→U is said to be graph dominated on A if (θ,Sθ)∈E(G) for all θ∈A.
Theorem 3.2. Let (U,dlb) be a complete rectangular b -metric space endowed with a graph G, {θn} be a Picard sequence and S:U→U be a graph dominated mapping on {θn}. Suppose that the following hold:
(i) there exists δb∈ϖb such that
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)), | (3.1) |
for all θ,ν∈{θn} and (θn,ν)∈E(G). Then (θn,θn+1)∈E(G) and {θn} converges to θ∗. Also, if (3.1) holds for θ∗ and (θn,θ∗)∈E(G) for all n∈N∪{0}, then S has a fixed point θ∗ in U.
Proof. Define α:U×U→[0,+∞) by
α(θ,ν)={1, ifθ,ν∈U, (θ,ν)∈E(G)14, otherwise. |
Since S is a graph dominated on {θn}, for θ∈{θn},(θ,Sθ)∈E(G). This implies that α(θ,Sθ)=1 for all θ∈{θn}. So S:U→U is an α-dominated mapping on {θn}. Moreover, inequality (3.1) can be written as
dlb(Sθ,Sν)≤δb(Dlb(θ,ν)), |
for all elements θ,ν in {θn} with α(θ,ν)≥1. Then, by Theorem 2.1, {θn} converges to θ∗∈U. Now, (θn,θ∗)∈E(G) implies that α(θn,θ∗)≥1. So all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, S has a fixed point θ∗ in U.
The authors would like to thank the Editor, the Associate Editor and the anonymous referees for sparing their valuable time for reviewing this article. The thoughtful comments of reviewers are very useful to improve and modify this article.
The authors declare that they have no competing interests.
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